1. Field of the Invention
The present invention relates to an image processing apparatus and method for analyzing and synthesizing signals of a dynamic image etc. utilized in fields of image recording, communications, etc.
2. Description of the Related Art
In recent years, as an approach for the analysis and synthesis of image signals etc., for example, a method has been reported in which note is taken of the edges of the signals, such as the boundary between the object and background in the input signals, and original signals are synthesized from the information at the edges of the signals.
For example, the image processing apparatus and method disclosed in "Second Generation Compact Image Coding With Wavelet" by Jacques Froment and Stephane Mallet, Wavelets--Tutorial in Theory and Applications, II: 655 to 678, 1991 (herein after, reference document 1) is one of the above-described methods. An explanation will be made below of this method.
FIGS. 1A to 1F are views showing one-dimensional signals f(x).
First, a one-dimensional signal f(x) as shown in FIGS. 1A and 1D is assumed. In this example, analysis is first carried out on the signal f(x) using the multiple resolution method.
Here, the "multiple resolution method" means a method of analyzing a signal using a filter having a number of different resolutions.
The characteristic (function) of the analyzing filter used in the analysis using this multiple resolution method is expressed by .phi..sub.a (x). This function .phi..sub.a (x) is expressed by the following equation in a frequency domain (region) .omega.. ##EQU1##
Note, the "a" of the filter function .phi..sub.a (x) expresses the scale in the multiple resolution. Also, the "i" in equation 1 expresses an imaginary number. Also, (hat) expresses a Fourier transformation of the function given this symbol (the same below).
This is similarly the result of differentiation of the function .theta.(x) in the original region x of the function expressed by the following equation in the frequency domain .omega.. ##EQU2##
The result by convolution computation of the filter function .phi.(px) and the signal f(x) is expressed as a function W.sub.a f(x). In the above reference document 1, it was indicated that an original signal can be approximated using just the filter output W.sub.a f(a x n) of the position (a x n) (note, n denotes a natural number) giving the maximum value of this amplitude .vertline.W.sub.a f(x).vertline..
Before explaining the restoration of the image signal from the maximum value, an explanation will be made concerning the restoration of the original signal f(x) from W.sub.af (x).
The method mentioned here is called a "wavelet" method and has been studied in depth in recent years.
Where the analysis function is expressed by the following equation: EQU .phi..sub.j (.omega.)=-i.omega.[sin(2.sup.-j.omega./4) / (2.sup.-j.omega./4)].sup.3 (3)
a filter function O,*(x) used for the synthesis is expressed by the following equation: ##EQU3##
Here, the line (-) drawn over the function expresses a complex conjugate. Also, the scale "a" is selected so that "a" becomes equal to 2j. Note, j=j1, . . . , jJ.
Here, summarizing the analysis and synthesis, if the original signal is defined as f.sub.0 (x), the following is established: EQU S.sub.jJ (x)=.theta..sub.jJ (x) * f.sub.0 (x) (5) EQU f(x)=f.sub.0 (x)-S.sub.jJ (x) (6)
and EQU W.sub.j f(x)=.phi..sub.j (x) * f(x) (7)
where, j=j1, . . . , J is analyzed, and synthesis is performed by: ##EQU4## EQU f.sub.0 (x0=f(x)+S.sub.jJ (x) (9)
In this way, the original signal f.sub.0 (x) can be restored from Wjf(x) and S(jJ) (x).
Accordingly, by interpolating 'W.sub.j f(x) from the previously mentioned W.sub.j f(xn), W.sub.j f(x) is approximated, and an approximation of f can be obtained by the abovementioned inverse transformation. The method of interpolation is one in which when it is assumed that the function obtained by interpolation is W.sub.j f(x), by using the fact that a function such as: EQU e(x)=W.sub.j f(x)-W.sub.j f(x) (10)
is transformed to a form such as: EQU e(x)=.alpha.exp(Ax)+.beta.exp(-Ax) (11)
and it passes through two points jxn and jx(n+1), u1 and v1 in the above equation are found from: EQU e(.sub.j x.sub.n)=W.sub.j f(.sub.j x.sub.n)-W.sub.j f(.sub.j x.sub.n) (12) EQU e(.sub.j x.sub.n+1)=W.sub.j f(.sub.j x.sub.n+1)-W.sub.j f(.sub.j x.sub.n+1) (13)
they are substituted in equation 11, which is added to the interpolation function, whereby the interpolation function is renewed.
In this, an estimated value f'(x) of the signal f(x) is found using equation 8, which is re-defined as W.sub.j f(x) using equation 7 again. By repeating this, f'(x) is renewed to restore f(x).
Finally, f.sub.0 (x) is obtained using equation 9.
The above-mentioned method of signal analysis and synthesis has the problems as mentioned below.
For example, where the input signal f.sub.0 (x) has a waveform as shown in FIG. 1A, the result of analysis for this signal by an analyzing filter of the type performing a first order differentiation becomes a waveform as shown in FIG. 1B, and the maximum point of absolute value of amplitude of waveform of the result of analysis exists at the two points indicated by (a) and (b) in FIG. 1B.
On the other hand, the result of analysis by the analyzing filter of the type performing a second order differentiation on the same signal f.sub.0 (x) becomes a waveform as shown in FIG. 1C, and the maximum point of the absolute value of amplitude of the waveform as the result of analysis exists at only one point indicated by (c) in FIG. 1C, and therefore the amount of data among the results of analysis which should be stored is smaller than that of the case using the analyzing filter of the type performing a first order differentiation.
However, where the input signal f.sub.0 (x) has a waveform as shown in FIG. 1D, the result of analysis by the analyzing filter of the type performing a first order differentiation for this signal becomes a waveform as shown in FIG. 1E, and the maximum point of absolute value of amplitude of the waveform as the result of analysis exists at two points indicated by (d) and (e) in FIG. 1E.
On the other hand, the result of analysis by the analyzing filter of the type performing a second order differentiation for this signal f.sub.0 (x) becomes a waveform as shown in FIG. 1F, the maximum points of absolute value of amplitude of the waveform as the result of analysis become four points indicated by (f) to (i) in FIG. 1F, and the number of maximum value in the case of using the analyzing filter of the type performing a first order differentiation conversely becomes smaller in comparison with the case where the input signal has a waveform as shown in FIG. 1A.
Seeing this from the viewpoints of both of the description and storage of the information, the characteristic point of a signal where the input signal f.sub.0 (x) has a waveform as shown in FIG. 1A should be given by the value (peak) of the maximum point of the absolute value of amplitude of the filter output signal as shown in FIG. 1C using an analyzing filter of the type performing a second order differentiation.
On the other hand, the characteristic point of a signal where the input signal f.sub.0 (x) has a waveform as shown in FIG. 1D should be given by the peak of the filter output signal as shown in FIG. 1E using the analyzing filter of the type performing a first order differentiation.
Where the information is coded by the aforesaid method, when a specific analyzing filter is applied to all of the input signals, the amount of data of the result of analysis is not always minimized with respect to the input signal, and there is a problem in that a sufficient compression rate can not be obtained.
Other than the method of compression of the image mentioned above, a large number of image compression methods have been conceived of, but there is a demand that the image quality and compression rate be further enhanced.
So as to respond to this demand, an attempt has been reported in which consideration is given to the fact that for a two-dimensional image, human vision is sensitive to the edges at which the luminance abruptly changes and that portion is accurately reproduced.
Also, a dynamic image may consist of two-dimensional spatial signals arranged in the time direction, and thus it can be regarded as a three-dimensional spatial image. Human vision may identify objects moving in the time direction as it does with respect to non-time-varying images.
However, a particularly sensitive reaction to moving edges is observed in the nerve cells processing visual information. Accordingly, as with the fact that the detection of a spatial, or position defined edge is important for human perception of objects, the existence of a moving edge that is defined in terms of position versus time is important for detection of objects moving in a time direction.
Usually, in compression of images using the detection of movement, the movement is detected by a certain method, the time direction is detected using this, and the error is coded.
However, for a dynamic image, a method of image processing (compression) and synthesis (reproduction) for accurately reproducing an edge part of the image using the fact that human vision is sensitive with respect to an edge at which the luminance (brightness) abruptly changes has not yet been reported.